Question is:

if \(\displaystyle \vec{a}=3i-2j\) , \(\displaystyle \vec{b}=-4i+4j\) , and \(\displaystyle \vec{c}=6i+-9j\) express the following vectors in their simplest form:

(i)\(\displaystyle -5\vec{b}\)

(ii)\(\displaystyle 2\vec{a}-\frac{1}{2}\vec{c}\)

(iii)\(\displaystyle \frac{2}{3}\vec{a}-\frac{1}{2}\vec{b}-\frac{1}{4}\vec{c}\)

My Solution:

(i) \(\displaystyle -5\left(-4i+4j \right)=20i+\left(-20j \right)\)

(ii) \(\displaystyle 2(3i-2j)-\frac{1}{2}(6i+(-9)j)\)

\(\displaystyle = 6i-4j-3i+(-\frac{9}{2})j\)

\(\displaystyle = (6-3=3i) -4j-\frac{9}{2}\)

\(\displaystyle = -\frac{4}{1}-\frac{9}{2}\)

\(\displaystyle =-\frac{8}{2}-\frac{9}{2}\)

\(\displaystyle =-\frac{17}{2}\)

Answer? \(\displaystyle 3i-\frac{17}{2}j\)

(iii) \(\displaystyle \frac{2}{3}(3i-2j)-\frac{1}{2}(-4i+4j)-\frac{1}{4}(6i+(-9)j)\)

\(\displaystyle (2i-\frac{4}{3}j)-(-2i+2j)-(\frac{3}{2}i+(-\frac{9}{4}j)\)

\(\displaystyle 2i+2i-\frac{3}{2}i\)

\(\displaystyle = \frac{4}{1}i+\frac{3}{2}i\)

\(\displaystyle = \frac{8}{2}+\frac{3}{2}\)

\(\displaystyle =\frac{11}{2}i\)

\(\displaystyle -\frac{4}{3}j+\frac{2}{1}j-\frac{9}{4}j\)

\(\displaystyle = -\frac{16}{12}j+\frac{24}{12}j-\frac{27}{12}j\)

\(\displaystyle = -\frac{19}{12}j\)

Answer? \(\displaystyle \frac{11}{2}i-\frac{19}{12}j\)